ELECTRONIC RESEARCH ANNOUNCEMENTS
OF THE AMERICAN MATHEMATICAL SOCIETY
Volume 2, Number 1, August 1996
A NEW PROOF OF THE FOUR-COLOUR THEOREM
NEIL ROBERTSON, DANIEL P. SANDERS, PAUL SEYMOUR, AND ROBIN THOMAS
(Communicated by Ronald Graham)
Abstract. The four-colour theorem, that every loopless planar graph admits
a vertex-colouring with at most four different colours, was proved in 1976 by
Appel and Haken, using a computer. Here we announce another proof, still
using a computer, but simpler than Appel and Haken’s in several respects.
1. Introduction
For our purposes a graph G consists of a finite set V (G) of vertices, a finite
set E(G) of edges, and an incidence relation between them. Each edge is incident
with two vertices, called its ends. An edge whose ends are equal is called a loop.
A plane graph is a graph embedded in the plane (without crossings, in the usual
way). The four-colour theorem (briefly, the 4CT) asserts that every loopless plane
graph admits a 4-colouring, that is, a mapping c : V (G) → {0, 1, 2, 3} such that
c(u) 6= c(v) for every edge of G with ends u and v. This was conjectured by F.
Guthrie in 1852, and remained open until a proof was found by Appel and Haken
[3], [4], [5] in 1976.
Unfortunately, the proof by Appel and Haken (briefly, A&H) has not been fully
accepted. There has remained a certain amount of doubt about its validity, basically
for two reasons:
(i) part of the A&H proof uses a computer, and cannot be verified by hand, and
(ii) even the part of the proof that is supposed to be checked by hand is extraordinarily complicated and tedious, and as far as we know, no one has made a
complete independent check of it.
Reason (i) may be a necessary evil, but reason (ii) is more disturbing, particularly
since the 4CT has a history of incorrect “proofs”. So in 1993, mainly for our own
peace of mind, we resolved to convince ourselves somehow that the 4CT really was
true. We began by trying to read the A&H proof, but very soon gave this up. To
check that the members of their “unavoidable set” were all reducible would require
Received by the editors September 26, 1995.
1991 Mathematics Subject Classification. Primary 05C10, 05C15, 05C85.
Research partially performed under a consulting agreement with Bellcore, and partially supported by DIMACS Center, Rutgers University, New Brunswick, New Jersey 08903, USA.
Neil Robertson was partially supported by NSF under Grant No. DMS-8903132 and by ONR
under Grant No. N00014-92-J-1965.
Daniel P. Sanders was partially supported by DIMACS and by ONR under Grant No. N0001493-1-0325.
Robin Thomas was partially supported by NSF under Grant No. DMS-9303761 and by ONR
under Grant No. N00014-93-1-0325.
c
1996
N. Robertson, D. P. Sanders, P. Seymour, R. Thomas
17
18
NEIL ROBERTSON, DANIEL P. SANDERS, PAUL SEYMOUR, AND ROBIN THOMAS
a considerable amount of programming, and also would require us to input by hand
into the computer descriptions of 1478 graphs; and this was not even the part of
their proof that was most controversial. We decided it would be easier, and more
fun, to make up our own proof, using the same general approach as A&H. So we
did; it was a year’s work, but we were able to convince ourselves that the 4CT is
true and provable by this approach [14]. In addition, our proof turned out to be
simpler than that of A&H in several respects.
The basic idea of the proof is the same as that of A&H. We exhibit a set of
“configurations”; in our case there are 633 of them. We prove that none of these
configurations can appear in a minimal counterexample to the 4CT, because if one
appeared, it could be replaced by something smaller, to make a smaller counterexample to the 4CT (this is called proving “reducibility”; here we are doing exactly
what A&H and several other authors did — for instance, [1], [2], [5], [9]). But every minimal counterexample is an “internally 6-connected triangulation” (defined
later), and in the second part of the proof we prove that at least one of the 633
configurations appears in every internally 6-connected triangulation. (This is called
proving “unavoidability”.) Consequently, there is no minimal counterexample, and
so the 4CT is true. Where our method differs from A&H is in how we prove
unavoidability.
Some aspects of this difference are: we confirm Heesch’s conjecture that one
can prove unavoidability of some reducible set without looking beyond the second
neighbourhoods of “overcharged” vertices; consequently, we avoid the problems
with configurations that “wrap around and meet themselves”, that were a major
source of complications for A&H; our unavoidable set has size about half that of
the A&H set; our “discharging procedure” for proving unavoidability (derived from
an elegant method of Mayer [13]) only involves 32 discharging rules, instead of the
487 of A&H; and we obtain a quadratic time algorithm to find a 4-colouring of a
planar graph, instead of the quartic algorithm of A&H.
Our proof is also somewhat easier to check, because we replace the mammoth
hand-checking of unavoidability that A&H required, by another mammoth handcheckable proof, but this time written formally so that, if desired, it can be read
and checked by a computer in a few minutes. We are making the necessary programs and data available to the public for checking via “anonymous ftp” from
ftp.math.gatech.edu in the directory pub/users/thomas/four.
The paper is organized as follows. In section 2 we introduce the concept of a
configuration, and state the two main results. In section 3 we discuss reducibility,
and in section 4 unavoidability. In section 5 we briefly discuss a quadratic algorithm
to 4–colour planar graphs, and in section 6 we make some concluding remarks.
2. Configurations
The degree of a vertex v of a graph G is the number of edges incident with it,
and is denoted by dG (v). A region of a plane graph is a triangle if it is incident with
precisely three edges. A plane graph is a triangulation if it is connected, loopless
and every region is a triangle. Thus, a triangulation can have parallel edges, but
no circuit of length two bounds a region.
A minimal counterexample means a plane graph G that is not 4-colourable, such
that every plane graph G0 with |V (G0 )| + |E(G0 )| < |V (G)| + |E(G)| is 4-colourable.
To prove the 4CT we must show that there is no minimal counterexample. It is
A NEW PROOF OF THE FOUR-COLOUR THEOREM
19
Figure 1. A set of configurations.
easy to show that every minimal counterexample is a triangulation, and almost is
6-connected. More precisely, let us say a graph G is internally 6-connected if G
has at least six vertices, and for every set X of vertices of G such that the graph
G\X obtained from G by deleting X is disconnected, either |X| ≥ 6, or |X| = 5
and G\X has exactly two components, one of which has exactly one vertex. Thus
every vertex of an internally 6-connected graph has degree at least five. We have
(see, for example, [7])
(2.1) Every minimal counterexample is an internally 6-connected triangulation.
A near-triangulation is a non-null connected loopless plane graph G such that
every finite region is a triangle. A configuration K consists of a near-triangulation
G(K) and a map γK : V (G(K)) → Z with the following properties:
(i) for every vertex v, G(K)\v has at most two components, and if there are two,
then γK (v) = d(v) + 2,
(ii) for every vertex v, if v is not incident with the infinite region, then γK (v) =
d(v), and otherwise γK (v) > d(v); and in either case γK (v) ≥ 5, P
(iii) K has ring-size ≥ 2, where the ring-size of K is defined to be (γK (v) −
v
d(v) − 1), summed over all vertices v incident with the infinite region such
that G(K)\v is connected.
Two configurations are isomorphic if there is a homeomorphism of the plane
mapping G(K) to G(L) and γK to γL . In [14] we exhibit a set of 633 configurations
that are essential to our proof. The set is also available in electronic form by
“anonymous ftp” as described earlier. Due to space limitations the full set cannot
be described here, but in Figure 1 we show a small subset that will be referred
to later in the paper. In Figure 1 we use a convention introduced by Heesch [11].
The shapes of vertices indicate the value of γK (v) as follows: A solid black circle
means γK (v) = 5, a dot (or what appears in the picture as no symbol at all) means
γK (v) = 6, a hollow circle means γK (v) = 7, and a hollow square means γK (v) = 8.
(We will not need vertices v with γK (v) > 8 in this paper.)
20
NEIL ROBERTSON, DANIEL P. SANDERS, PAUL SEYMOUR, AND ROBIN THOMAS
Any configuration isomorphic to one of the 633 configurations exhibited in [14]
is called a good configuration. Let T be a triangulation. A configuration K appears
in T if G(K) is an induced subgraph of T , every finite region of G(K) is a region of
T , and γK (v) = dT (v) for every vertex v ∈ V (G(K)). We prove the following two
statements.
(2.2) If T is a minimal counterexample, then no good configuration appears in T .
(2.3) For every internally 6-connected triangulation T , some good configuration
appears in T .
From (2.1), (2.2) and (2.3) it follows that no minimal counterexample exists, and
so the 4CT is true. We shall discuss (2.2) in section 3, and (2.3) in section 4. The
first proof needs a computer. The second can be checked by hand in a few months,
or, using a computer, it can be verified in a few minutes.
3. Reducibility
By a circuit we mean a non-null connected graph in which every vertex has
degree two. Let K be a configuration. A near-triangulation S is a free completion
of K with ring R if
(i) R is an induced circuit of S, and bounds the infinite region of S,
(ii) G(K) is an induced subgraph of S, G(K) = S\V (R), every finite region of
G(K) is a finite region of S, and the infinite region of G(K) includes R and
the infinite region of S, and
(iii) every vertex v of S not in V (R) has degree γK (v) in S.
It is easy to check that every configuration has a free completion. (This is
where we use the condition that ring-size ≥ 2 in the definition of a configuration
— the ring-size is actually the length of the ring in the free completion, as the
reader may verify.) Moreover, if S1 , S2 are two free completions of K, there is a
homeomorphism of the plane fixing G(K) pointwise and mapping S1 to S2 . (This
is where condition (i) in the definition of a configuration is used.) Thus, there is
essentially only one free completion, and so we may speak of “the” free completion
without serious ambiguity.
Let R be a circuit. There is a concept of “consistency” of a set of 4-colourings of
R which essentially dates back to Kempe [12] and Birkhoff [7]. We will not define
this concept here, but instead mention the properties we need. They are:
(i) the empty set is consistent,
(ii) the union of any two consistent sets is consistent,
(iii) if T is a triangulation, R is a circuit of T , T 0 is the near-triangulation obtained
from T by deleting all vertices of T that belong to the open disc bounded by
R, and C is the set of restrictions of all 4-colourings of T 0 to R, then C is
consistent, and
(iv) consistency is sufficiently restrictive to permit enough reducible configurations
(defined later).
Let S be the free completion of a configuration K with ring R. Let C ∗ be the set
of all 4-colourings of R, and let C be the set of all restrictions to V (R) of 4-colourings
of S. Let C 0 be the maximal consistent subset of C ∗ − C. (This is well defined by (i)
and (ii).) The configuration K is said to be D-reducible if C 0 = ∅, and is said to be
C-reducible if there exists a near-triangulation S 0 , called a reducer, obtained from
A NEW PROOF OF THE FOUR-COLOUR THEOREM
21
S by replacing G(K) by a smaller graph (and possibly identifying some vertices
of R) such that no member of C 0 is the restriction to V (R) of a 4-colouring of S 0 .
It is reasonably easy to show that no D-reducible configuration can appear in a
minimal counterexample. (Notice that the appearance of a configuration in T does
not imply that the free completion is a subgraph of T , but this difficulty is easy to
take care of.) A more serious problem arises with C-reduction. There the natural
proof that no C-reducible configuration can appear in a minimal counterexample
would “replace” S by S 0 in T , but how do we know that the resulting graph is
loopless? Let us call a reducer safe if the above process results in a triangulation
for every internally 6-connected triangulation T . Checking safety was a major
source of complications for A&H, and one advantage of our new proof is that we
are able to avoid these difficulties. The way we handle this problem is that we only
consider reducers that are obtained from K by contracting at most four edges, for
which the safety check is easy.
In order to prove (2.2) we show that
(3.1) Each of the good 633 configurations is either D-reducible or C-reducible with
a safe reducer.
Theorem (3.1) is proved by means of a computer program that is available to
the public for examination. In fact, we used two independent programs that also
computed some numerical invariants that could be compared to double-check our
results.
4. Unavoidability
A configuration K appears in a configuration L if G(K) is an induced subgraph
of G(L), every finite region of K is a finite region of L (and hence the infinite region
of K includes the infinite region of L), and γK (v) = γL (v) for every v ∈ V (G(K)).
Let T be an internally 6-connected triangulation. For a vertex v of T we define
the vicinity of v in T to be the configuration K with G(K) the subgraph of T
induced by all vertices u of T such that there is a path P in T with ends u and v
with at most three vertices, and such that the interior vertex of P (if there is one)
has degree at most eight, and γK (v) = dT (v) for v ∈ V (G(K)). To prove (2.3)
we show that a good configuration appears in the vicinity of some vertex. We now
explain how we locate such a vertex.
A rule is a 6-tuple (G, α, , r, s, t), where G is a near-triangulation, α and are
mappings from V (G) to {5, 6, 7, 8} and {−, 0, +}, respectively, r > 0 is an integer,
and s and t are distinct adjacent vertices of G. Figure 2 describes a set R of 32
rules as follows. The mapping α is described using the same conventions as for
Figure 1, is described by placing an appropriate sign next to the corresponding
vertex (“0” is omitted), r = 2 for the first rule and r = 1 for all the other rules,
and s, t ∈ V (G) are such that the unique directed edge of the rule is directed from
s to t. There is some system in the first nine rules, but the other rules were chosen
by trial and error, and have no particular plan or pattern.
A pass P is a quadruple (K, r, s, t), where K is a configuration, r > 0 is an
integer and s, t ∈ V (G(K)) such that there exists a rule R = (G, α, , r, s0 , t0 ) ∈ R
and a homeomorphism h of the plane mapping G(K) onto G such that
(i) γK (v) ≤ α(h(v)) for every v ∈ V (G(K)) with (h(v)) ∈ {−, 0},
(ii) γK (v) ≥ α(h(v)) for every v ∈ V (G(K)) with (h(v)) ∈ {0, +}, and
22
NEIL ROBERTSON, DANIEL P. SANDERS, PAUL SEYMOUR, AND ROBIN THOMAS
Figure 2. The set R of rules.
(iii) h(s) = s0 and h(t) = t0 .
We say that P obeys rule R. It is straightforward to verify that every pass obeys
exactly one rule. We write r(P ) = r, s(P ) = s, t(P ) = t, and K(P ) = K. We call r
the value of the pass, s its source, and t its sink. A pass P appears in a triangulation
T if K(P ) appears in T . A pass P appears in a configuration L if K(P ) appears in
A NEW PROOF OF THE FOUR-COLOUR THEOREM
23
L. If v is a vertex of a triangulation T , we define ΘT (v) to be
X
10(6 − dT (v)) +
(r(P ) : P appears in T , t(P ) = v)
X
−
(r(P ) : P appears in T , s(P ) = v) .
We remark that ΘT (v) depends only on the vicinity of v in T .
It follows from Euler’s formula that
P
(4.1) For every triangulation T , v∈V (T ) ΘT (v) = 120. In particular, there is a
vertex v of T with ΘT (v) > 0.
Thus in order to prove (2.3) it suffices to prove
(4.2) If v is a vertex of an internally 6-connected triangulation T with ΘT (v) > 0,
then a good configuration appears in the vicinity of v in T .
Let C1 , C2 , . . . , C17 be the configurations pictured in Figure 1. The following
can be shown without the use of a computer.
(4.3) If v is a vertex of an internally 6-connected triangulation T with ΘT (v) > 0,
and either dT (v) ≥ 12 or dT (v) ≤ 6, then a configuration isomorphic to one of
C1 , C2 , . . . , C17 appears in the vicinity of v in T .
Sketch of proof. Assume that dT (v) ≥ 12, and suppose that no such configuration
appears. Let dT (v) = d and let D be the set of neighbours of v. For each u ∈ D,
let R(u) be the sum of r(P ) over all passes P appearing in T with source u and
sink v. It can be shown that R(u) ≤P5 for every u ∈ D (we omit the details — this
is just a finite case analysis). Thus u∈D R(u) ≤ 5d, and hence
ΘT (v) = 10(6 − d) +
X
R(u) ≤ 10(6 − d) + 5d = 60 − 5d ≤ 0 ,
u∈D
a contradiction. This completes the proof in the case when dT (v) ≥ 12. We omit
the other part, because, again, it is just a finite case analysis.
Thus in order to prove (4.2) and hence (2.3) it remains to prove the following.
(4.4) If v is a vertex of an internally 6-connected triangulation T with ΘT (v) > 0
and dT (v) = 7, 8, 9, 10 or 11, then a good configuration appears in the vicinity of v
in T .
For each of the five cases, we have a proof. Unfortunately they are very long
(altogether about 13,000 lines, and a large proportion of the lines take some thought
to verify), and so cannot be included in a journal article. Moreover, although any
line of the proofs can be checked by hand, the proofs themselves are not “really”
checkable by hand because of their length. We therefore wrote the proofs so that
they are machine-readable, and in fact a computer can check these proofs in a
few minutes. More information is given in [14, Section 7], and full details can be
obtained by “anonymous ftp” as described earlier. Alternatively, one can write a
computer program to check (4.4) directly, for it is easily seen to be a finite problem.
24
NEIL ROBERTSON, DANIEL P. SANDERS, PAUL SEYMOUR, AND ROBIN THOMAS
5. An algorithm
Our proof gives a quadratic algorithm to four-colour planar graphs, which is an
improvement over the quartic algorithm of A&H. Let us clarify a possible confusion
here. It might appear that there is an “obvious” quadratic algorithm as follows.
Given an input plane graph G on n vertices (which may as well be assumed to be a
triangulation) we find a good configuration appearing in G, replace G by a smaller
graph G0 , and apply recursion. A reducible configuration can be found in linear
time, and a 4-colouring of G can be constructed from a 4-colouring of G0 in linear
time. Since there are at most n recursive steps, the overall running time would be
quadratic.
The problem is that (2.3) only guarantees the appearance of a good configuration
if G is an internally 6-connected triangulation, and yet even if we started with a
highly connected graph, the connectivity is bound to drop during the recursion.
However, (4.2) permits us to quickly find either a good configuration appearing
in G, in which case we proceed as suggested above, or a set X of vertices of G
violating the definition of internal 6-connection, in which case we apply recursion
to two carefully selected subgraphs of G, and then obtain a 4-colouring of G by
piecing together 4-colourings of the two subgraphs. We refer to [14] for further
details.
6. Discussion
This concludes our discussion of the proof of the 4CT and the associated algorithm. A full account can be found in [14] except for the proofs of (3.1) and (4.4).
Those two theorems are just stated in [14] as having been proved by a computer.
Verifying (3.1) takes about 3 hours on a Sun Sparc 20 workstation, and (4.4) takes
about 20 minutes altogether. Both need less than one megabyte of RAM. The
complete programs as well as documenation for them [15], [16] can be obtained
for examination by “anonymous ftp” as described earlier. Thus we are making it
possible for other scientists to verify all steps in our proof, including the computer
programs and data.
We should mention that both our programs use only integer arithmetic, and
so we need not be concerned with round-off errors and similar dangers of floating
point arithmetic. However, an argument can be made that our “proof” is not a
proof in the traditional sense, because it contains steps that can never be verified
by humans. In particular, we have not proved the correctness of the compiler we
compiled our programs on, nor have we proved the infallibility of the hardware
we ran our programs on. These have to be taken on faith, and are conceivably a
source of error. However, from a practical point of view, the chance of a computer
error that appears consistently in exactly the same way on all runs of our programs
on all the compilers under all the operating systems that our programs run on is
infinitesimally small compared to the chance of a human error during the same
amount of case-checking. Apart from this hypothetical possibility of a computer
consistently giving an incorrect answer, the rest of our proof can be verified in the
same way as traditional mathematical proofs. We concede, however, that verifying
a computer program is much more difficult than checking a mathematical proof of
the same length.
A NEW PROOF OF THE FOUR-COLOUR THEOREM
25
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Department of Mathematics, Ohio State University, 231 W. 18th Ave., Columbus,
Ohio 43210, USA
E-mail address: [email protected]
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332,
USA
E-mail address: [email protected]
Bellcore, 445 South Street, Morristown, New Jersey 07960, USA
E-mail address: [email protected]
School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332,
USA
E-mail address: [email protected]